Stability of monotone vorticities in the half cylinder and infinite perimeter growth for patches

Abstract

We consider the incompressible Euler equations in the half cylinder R>0 × T. In this domain, any vorticity which is independent of x2 defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted L 1 norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in x1. This includes stability of cylindrical patches {x1 < α}, α > 0. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time.